reference for citing the concept of vector-valued test functions?

[Simon]

Dear all,

is the deal.ii implementation of vector-valued test functions as described here  also described in an article, paper, ... or is this a "deal.ii thing"?

I would like to make a reference in a publication where the concept of vector-valued test functions is explained in more detail, but do not want to cite a webpage.

Thank you!

[Wolfgang Bangerth]

On 3/16/23 06:46, Simon wrote:
>
> is the deal.ii implementation of vector-valued test functions as described
> here

> <https://nam10.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.dealii.org%2Fcurrent%2Fdoxygen%2Fdeal.II%2Fgroup__vector__valued.html&data=05%7C01%7CWolfgang.Bangerth%40colostate.edu%7C95495b1c1a8e4d8eb6a108db261c7e45%7Cafb58802ff7a4bb1ab21367ff2ecfc8b%7C0%7C0%7C638145676051760102%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=vQVSQIlssdYtwa%2BsT40OyBVta8uXZ3%2FJppPEiy4F%2BoE%3D&reserved=0>  also described in an article, paper, ... or is this a "deal.ii thing"?
>

Simon:
I think it is the way it needs to be done, not just in deal.II. Take a look,
for example, at step-20
https://www.dealii.org/current/doxygen/deal.II/step_20.html#Formulationweakformanddiscreteproblem
where we define the bilinear form as
a({u,p},{v,q})
which is how you will see it in many publications. The point is that the
bilinear form must result in a scalar number, and you get that by
dot-multiplying the several equations of the PDE system by a vector and then
integrating over the domain.

In other words, while I can't point at a specific publication or book, this is
how it is commonly done. I don't think you need to cite anything.

Best
W.

--
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Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/